A manufacturer of a product sold with a warranty normally provides service, replacement or, in some cases, a refund to a customer when the product becomes defective within a warranty period. Such service, replacement or a refund under the warranty incurs a warranty cost.
Warranties come in different types. The most common type is the so-called "individual warranty," whereby each product unit is individually warranted for a specified period. By contrast, a cumulative warranty covers a batch of units as a whole, and the warranty period applies to the entire batch. Such a cumulative warranty is normally used to cover reliability systems wherein system components are readily replaceable with spare parts.
Thus, for a product sold with a warranty, quality control is important in the manufacturing process to reduce the warranty cost. To this end, before units of the product generally grouped in batches are delivered to customers, the manufacturer normally performs an inspection to ensure their quality.
A typical inspection is conducted based on a known defect ratio .THETA., defined as a proportion of defective units within a batch. This defect ratio may be derived from historical data and/or knowledge about the manufacturing process. Taking into account cost factors such as inspection, repair and warranty costs, the manufacturer makes a decision whether to inspect the whole batch or not to inspect at all. It is apparent that the decision would favor an inspection when .THETA. is relatively high and the inspection and repair costs are relatively low. According to C. Tapiero et al., "Quality Control and Product Servicing: A Decision Framework," European Journal Operational Research, vol. 39, 1989, pp. 261-273, when .THETA. is deterministically known for the batch under inspection, the 0 or 100% inspection rule is optimal in the case where an individual warranty for example is extended. However, in practice the .theta. value derived from the historical data may bear slight relevance to each particular batch, thus leading to a faulty decision whether to inspect or not.
What complicates the matter is that the defect ratio .THETA. is oftentimes a random variable and thus undeterministically known. That is, .THETA. may only be realized as a statistical distribution, in which case Tapiero suggests a partial inspection. A typical partial inspection calls for examination of sample units from a batch. Depending on the number of defective units identified within the sample units, the manufacturer decides whether to reject or accept the entire batch. However, the success of this procedure requires an impeccable choice of the sample size. If the sample size is too small, the procedure becomes unreliable. On the other hand, if the size is too large, the procedure becomes unjustifiably expensive.
Accordingly, it is desirable to have a cost-effective methodology for inspection of finished products sold with warranties even though their defect ratios are undeterministic.